The Natural Sciences and Mathematics || Mathematics in the Elementary Grades

Mathematics in the Elementary GradesAuthor(s): E. Glenadine Gibb and H. Van EngenSource: Review of Educational Research, Vol. 27, No. 4, The Natural Sciences and Mathematics(Oct., 1957), pp. 329-342Published by: American Educational Research AssociationStable URL: http://www.jstor.org/stable/1169238 .

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CHAPTER II

Mathematics in the Elementary Grades

E. GLENADINE GIBB and H. VAN ENGEN

RESEARCH related to the arithmetic curriculum has had much influence on teaching procedures, content of arithmetic at various grade levels, organization of instructional groups, and programs of curriculum improvement. Certain research findings, once widely accepted, were re- evaluated during the period under review.

Summaries and Bibliographies

Six years ago Burch and Moser (16) reviewed in a similar issue of this publication the research of the previous three-year period. Since that time there have been several summaries and interpretations of research. In annual annotated bibliographies Hartung (48) presented selected references pertaining to instruction in arithmetic in the elementary school. Wrightstone (112) summarized selected research studies that have in- fluenced instruction in arithmetic, noting that mature and successful teachers are far from being in complete agreement with or accepting current research findings and judgments of experts. Gibb (34) summarized experimental investigations which compared methods of teaching; she also prepared a selected annotated bibliography (35) for the classroom teacher interested in action research. In publications sponsored by the Association for Supervision and Curriculum Development and the Ameri- can Educational Research Association, both departments of the National Education Association, Glennon and Hunnicutt (37) and Morton (70) examined research for its implications for the classroom teacher. Pikal (81) reviewed research related to methods of teaching, to concepts learned, and to the use of arithmetic outside the school in the teaching of arithmetic in the upper elementary grades. Weaver (106) summarized the research related to elementary-school mathematics for the years 1951-1956, pre- senting an annotated bibliography and some concluding observations concerning research during that period of time.

Methods of Instruction

No longer is research limited to emphasis on 100-percent accuracy in computation such as that summarized by Wilson (108) in a study of the mechanics of computation. In a review of research dealing with the effect of instructional procedures on achievement in fundamental operations in arithmetic, Hightower (51) suggested that superior instruction and motivation may be more important than the mechanics of the method used.

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Evidence continued to accumulate in support of the discovery method of teaching whereby children are encouraged to structure mathematical concepts. Dawson and Ruddell (21) reviewed four studies showing what is to be gained thru the "meaning method" of teaching arithmetic. As the result of the Los Angeles study, Miller (69) reported that in comparing the "meaning" and "rule" methods of teaching, the "meaning" method was more effective for the areas of (a) computation of fractions, (b) establishing retention, (c) processes of computation, and (d) the understanding of the principles of arithmetic needed to comprehend complex analyses in arithmetic.

In investigating the possibilities of teaching arithmetic by television, Strueve (96) found it opened a way for better understanding of present- day method, extended the influence of a superior teacher to many classrooms, and gave regular teachers more opportunity to help individual children. Articles in Emerging Practices in Mathematics Education (72) presented other classroom procedures.

Learning

Closely associated with research on methods of instruction were studies concerned not with the products of learning but with the process of learn-

ing. The Twenty-First Yearbook of the National Council of Teachers of Mathematics, Learning of Mathematics, (73) discussed important aspects of learning, theories of learning, motivation, formation of concepts, sen-

sory learning, language, drill, transfer of training, problem solving, individual differences, and improvement of instruction. Purdy and Kinney (86) presented a series of lessons as a sample of effective procedures that may be interpreted theoretically to describe a planned sequence for effective arithmetic learning. Buswell and Kersh (17) studied thought patterns in an effort to determine objectively general patterns of problem solving. They found the most striking characteristic to be the variety rather than the similarity in sequence of thinking. Van Engen and Gibb (103) found the general idea approach to learning more effective than a method of unit skills. Plank and Plank (82) identified the emotional components in arithmetical learning thru autobiographies of scientists, statesmen, and musicians in an effort to determine why many people have not learned mathematics. Baldwin (5) indicated an unexplored area of psychological research as he discussed number forms and mental imagery.

Abilities and Attitudes

Related to the thought patterns of learning are the ability to learn arithmetic and the attitude toward it. In a study of cognitive factors associated with progress of children in arithmetic, Barakat (6) found that innate intelligence appeared to contribute the largest part to mathematical

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attainment. He also found ample evidence for a specialized factor in mathematical ability, its components being mathematical reasoning and a

symbolic factor allied to mechanical computation and rote memory. Emo- tional instability correlated most highly with inaccuracy of computation, and lack of industry with inefficiency in mathematical reasoning.

An attitude scale constructed by Dutton (24) was used to estimate general attitude toward arithmetic. Grades V and VI were the most crucial in developing attitudes. Lynn (63) found a tendency for anxiety to be associated with good reading and poor arithmetic.

McAllister (64) attempted to find a relation between degree of devel-

opment of arithmetical concepts and intelligence, attention span, and ability to do both formal and functional arithmetic. Poffenberger and Norton

(83) examined factors affecting pupils' attitudes toward arithmetic. It appeared that parents and teachers were in the main responsible for pupils' positive or negative attitudes.

Concepts of Children

Martin (67) studied the spontaneous reactions of young children, con-

firming findings of previous studies that with advance in age, young children increase in their ability to handle concepts of number, size, and

quantity as measured in formal test situations. It was noted that a child who at first viewed collections of objects as a whole in terms of a single characteristic later readily identified specific objects and subgroups of

objects. In another study of number concept, Ilg and Ames (56) found the errors that children made to be the best clues to a child's stage of

development. Emphasis should then be placed on kinds of errors made, not on right and wrong answers. In investigating the development of children's understanding in the four fundamental processes, Deans (23) found that children dropped less mature responses in favor of more mature processes.

In a descriptive study of responses to nine problems, Gunderson (40) attempted to discover thought patterns of young children in solving multi-

plication and division problems before they had been taught the processes, directing attention to the need for readiness material for multiplication and division. The study indicated that measurement division was more easily understood and applied than was partitive division and that

probably division as a process should be introduced thru measurement

problems. Springer (95) designed a study to ascertain what children know about

the appearance and operation of a clock prior to formal school instruction. Trends in drawings showed that maturation of ideas and abilities to

express those ideas were enhanced by an increase in incidental experiences and observations. Piaget (80) described some experiments which indicated the essential idea of number development in children; he suggested that

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the historical development of geometry is reversed for children. Pratt and others (84) studied children's abilities to designate magnitude of indeterminate numbers as few, hardly, scarcely, several, some, lot, and many. Priore (85) tested children entering first grade to discover their knowledge of arithmetic. Flournoy (29) examined children's thought processes in higher-decade addition. She concluded that unless children were helped to establish a pattern of thinking which enabled them to perform higher-decade addition almost as if in one complete thought, they were likely to devise time-consuming and perhaps inaccurate ways of adding a column of figures. In a study of children's thought processes in subtraction, Gibb (33, 36) found that children responded to the action of the problem situation in solving the problem. She concluded that this caused difficulty in various additive and comparative situations which are com- monly processed by subtraction.

Number

Dawson (20), proposing complexity as a critical factor in apprehension of number as a group, found that the use of complicated pictures tended to produce counting, not grouping, and hence did not assist the learner in developing ability to recognize number groups. Woods (110) sought to discover whether children's arithmetical ability would be improved if they were taught to apprehend as groups certain lines in straight-line patterns. He found a low positive correlation between ability to group and general achievement in arithmetic skills. No significant improvement in arithmetic achievement resulted from special training in apprehending groups.

Operations

Thyne (98) made inquiry into the nature of error in the addition facts, summarizing types of errors and conditions of occurrence and concluding that orders of difficulty were of no real value. In a study of subtraction Weaver (107) found that it would be advantageous for all children to begin with a common method of compound subtraction-decomposition -and then follow different procedures. Meddleton (68) compared the results arising from the systematic practice of carefully compiled and graded number combinations, with results from the ordinary "laissez faire" procedures of presentation and practice. Conclusions were that children made more arithmetical progress with systematic revision sheets, that organized material was of slightly more benefit to the children in poorer socioeconomic areas, and that more improvement was made in subtraction and division than in multiplication and addition. Crumley (19) found that regardless of the method of teaching subtraction children had almost a universal understanding of subtraction as a "take-away" process. Rheins and Rheins (87) tried to determine whether the method of decom-

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position, or the method of equal addition, was superior in speed and accuracy in compound subtraction of whole numbers five years after the process had been taught. Results of their research did not favor one method strongly. Bebb (7) compared a classroom method of teaching basic multiplication facts with a tachistoscopic presentation, finding the latter to be only an interesting and varied method of learning. Similar findings by Phillips (79) indicated that tachistoscopic methods are effective but no more effective than the ordinary workbook method of practice.

Brownell (13) reported that children often found division difficult be- cause they had inadequate mastery of skills and facts basic to success in learning. He (14) believed it better to have mastery of subskills in division before going on with more problems in division. Dawson and Ruddell (22) compared the effectiveness of common textbook practices in introducing the division of whole numbers with an experimental procedure based on a subtractive approach and a greatly expanded use of visual- ization devices. These authors found higher achievement, greater retention, and increased ability to solve examples in a new situation where the experimental method was used. In another study Van Engen and Gibb (103) used visual material in the conventional and in the subtractive methods of division. They, too, found a greater ability to transfer learning to new situations among those using the subtractive method. However, achievement and retention were not different for the two methods. Benz (8) presented data showing how applications of two rules do not produce correct quotient figures. In a critical summary of research in estimating quotients, Hartung (47) brought to the reader's attention aspects of issues which have been largely neglected in estimating quotients. He considered advantages in using a trial quotient that is smaller than the true quotient and the relative simplicity of a "one rule" procedure. Thruout the litera- ture on the teaching of division, questions were raised regarding measurement and partition ideas in division. Moser (71) described some exploratory work in thinking of division as finding either the number of groups or the size of one group. Hill (52) discussed the use of measurement-type situations to introduce the idea of division.

Fractions

In a thoughtful discussion of fractions, Riess (88) observed that much valuable material on the history of fractions presented in scientific writ- ings has been neglected. She also suggested that fractions be approached from the standpoint of multiplication rather than division. Brooke (11) compared two methods of introducing division of fractions. Results indicated that the common denominator method might be more successful for use in introducing the idea of a whole number by a fraction if there were no remainders in the answer. Aftreth (2, 3, 4) determined the effect on the learning process of the identification and correction of typical errors embedded in exercises in addition and subtraction of fractions.

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Apparently, at least in the early stages, learning was not adversely affected. Results from Johnson's study (57) showed that it was easier to learn to use decimals than common denominators in addition and subtraction of fractions; a case was presented for teaching decimals first. Lankford and Pattishall (59) reported that allowing pupils much freedom and explora- tion in working with fractions was a sound teaching procedure. Hirsch (53) discovered that three test questions involving the division of fractions, altho essentially of the same type, did not represent the same degree of difficulty to pupils. One question was identical with examples in the text; children found it easier. The other two questions were phrased in terms only slightly different, and involved the same principles, but the pupils found them considerably more difficult.

Problem Solving Much has been written regarding formal analysis in problem solving.

Burch (15) found that children tended to score higher on tests not re- quiring them to go thru steps of formal analysis. Thinking more carefully about size, relationships, and dynamics of quantities described in each problem seemed a more profitable attack. Gunderson (39) described the variation in children's responses to problem situations and indicated that there was no one pattern for all children.

From a questionnaire investigation, Ullrich (99) found a difference of opinion among teachers, superintendents, and writers regarding the extent to which problem answers should be labeled. Much of what writers of textbooks know about acceptable answers is not always known by elementary-school teachers.

Van Engen (101) suggested a fertile field of research by noting that children learn to recognize what process to perform by visualizing what operations are indicated by the words in the problem.

Measures

According to McLatchy (66) preschool children have a better understanding of measures of time than of other conventional measures. Wilson and Cassell (109) summarized the research related to what should be taught in weights and measures. Cassell concluded that the experience factor should determine the teaching of weights and measures, for some commodities may be sold in different units in different sections of the country.

Mental Arithmetic

Hall (42) presented a case for his definition of mental arithmetic after summarizing meanings given to the term. In another investigation he (41) identified uses of mental arithmetic made by business people during one

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day's business affairs. Mechanical computing devices have not replaced the need for mental arithmetic.

Petty (77) studied the effectiveness of a non-pencil-and-paper method of solving verbal problems in arithmetic, finding that pupils trained in the method tended to use pencil and paper less.

Flournoy (31) found that oral arithmetic gave pupils greater confidence and understanding of the written word problem. Spitzer and Flournoy (94) gave attention to verbal problems.

Evaluation

Hannum (45) constructed and standardized a test of arithmetical skills and abilities for individual use with children from preschool to the second- grade level. This instrument should have value for an arithmetic readiness program in the primary grades. Wrightstone (111) reported the results of building a test of arithmetical meaning for Grades I and II. McKenzie (65) described procedures used to standardize Vernon's Graded Arith- metic Test.

In recent years there have been charges against the Three R's in our schools. Lanton (60) showed that a selected population of third- and fifth- grade children achieved higher scores on standardized educational tests than did a comparable group tested in the same way nearly two decades ago.

Harvey (49) reported the effectiveness of tests and materials in dis- covering errors in arithmetic caused by failure to understand zero. Frederiksen and Satter (32) illustrated several methods of test construction and validation for arithmetic which are uniquely or rarely employed. Bouchard (10) explored effects of selected variables upon performance of sixth-grade children in arithmetic. Superior performance was generally achieved by children in the experimental groups (a) who were given the correct answers to arithmetic exercises after each exercise was completed, and (b) who knew that children in other sixth grades were being given the same tests.

Individual Differences

A common problem in the teaching of arithmetic is the provision of equal opportunity for all pupils in a classroom. From a study of a retarded group and a normal group of boys, Hamza (44) identified intellectual, emotional, social, environmental, and pedagogical factors as contributing to retardation. Hoel (54) reported a two-weeks' experiment in clinical procedures in arithmetic; efforts were made to discover specific arithmetic weaknesses and to determine at what stage in the child's development the difficulties arose. Holmes and Harvey (55) concluded that the method used in grouping arithmetic classes in order to deal with individual dif-

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ferences was not of great importance; more crucial were the teacher's personal and professional qualities and methods of teaching.

Kliebhan (58) found that the principal differences between high and low achievers in problem solving were in ability to estimate answers and select steps in the process, and skill in fundamentals, computational ability, and quantitative understanding. No difference was found between attitudes of high and low achievers. Corle (18) found good problem-solvers superior to poor ones both in understanding of number relationships and in computation.

Another program for retarded children was described by Valentine (100), who found that they made four times as much progress in the special classrooms as could have been expected in a similar period of time under ordinary classroom conditions.

One of the most promising methods for dealing with individual differences, as described by Weaver (105), was differentiation of instruction. His summary of research findings should be of interest to those providing for individual differences in arithmetic instruction.

Aids to Learning

Thruout history, attention has been given to instructional aids. Adkins (1) described the use of physical devices thru the ages, giving particular attention to the tally, knotted cord, fingers, and abacus as means of record- ing, communicating, and computing with numbers. Edison (27) developed a 60-item test to be used in determining prospective teachers' understandings of the selection and use of such aids.

Fehr, McMeen, and Sobel (28) reported an experiment to measure the value of the use of computing machines in the learning of arithmetic; they found significant gains both in computation and reasoning on the part of the experimental group. Schott (90) showed that the abacus and adding machine were valuable tools for teaching arithmetic. Skinner (92) reported on the use of a mechanical device to provide controlled practice in developing skill in arithmetic.

Arithmetic Textbooks

Several studies were made of textbooks. Flournoy (30) examined six series published since 1950 and commented on the lack of agreement they showed regarding the teaching of higher-decade addition. Lee (62) analyzed the primary arithmetic textbooks of oriental countries and the United States, finding that instructional materials emphasized cultural as well as educational values. Harrison's investigation (46) dealt with semantic variation, ambiguity, and the introduction of new meanings as sources of difficulty. In a study of recently published arithmetic books, Glott (38) found that verbal matter in problem solving was more difficult

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than that in developmental and explanatory material. There were also extreme variations within textbooks.

In a study of seven series of arithmetic textbooks for Grades III thru VI by Bhargava (9) there was found to be high agreement on the sequence of common steps, but the scope of each topic differed as did the amount of space allotted; there were also many differences regarding the use of larger numbers and fractions.

Hensell (50) reviewed existing studies and proposed criteria for the evaluation of problem-solving content in arithmetic textbooks.

Teacher Education

In assessing teacher understandings and attitudes, Phillips (78) found that prospective teachers' reactions to mathematics were a result of method of presentation, opportunities for achievement, teacher's personality, and type of problem solved. It was again observed that negative reactions started in the intermediate grades. Orleans and Sperling (75) and Orleans and Wandt (76) presented further evidence of difficulties in the arithmetical understandings of teachers. Dutton (25) found that isolation of attitude toward a subject was helpful in stimulating prospective teachers to overcome unfavorable attitudes and to strengthen favorable feeling. In another study Dutton (26) reported on attitudes of prospective teachers as determined by an objective evaluation instrument. Sueltz (97) studied the kinds of questions asked by prospective teachers and invited experimental studies to answer them.

Perhaps a knowledge of real mathematics would help more than method in overcoming fears of the subject. From a study of views held by selected leaders in mathematics education, Snader (93) found that a minimum of six semester hours should be provided for a mathematics course designed for elementary-school teachers of arithmetic. Layton (61) reported that only one-fourth of 85 institutions of higher learning preparing elementary- school teachers required mathematics for admission to a teacher-education curriculum. The mean mathematics content requirement of the four-year elementary curriculum was approximately 1.6 semester hours, as compared with means of 4.3 semester hours for art and for geography, and 11.5 semester hours for English. Schaaf (89) presented some evidence that elementary-school teachers did not understand mathematics and made recommendations for a course in arithmetic for teacher-education institutions. Weaver (104) also presented evidence of teachers' lack of understandings vital to meaningful arithmetic instruction in the elementary school, and identified the responsibility of teacher-education institutions to improve arithmetic scholarship. In providing for the mathematics education of elementary-school teachers, Hamilton (43) assessed the effectiveness of some experimental material designed to acquaint prospective teachers with the nature of the number system.

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Future

As noted by Sherer (91), research in arithmetic has consisted mainly of brief cross-sectional investigations, inventories, short experimental studies, and test samplings of children's concepts, interests, abilities, and uses of arithmetic. Brown (12) predicted a new organization of arithmetic with basic skills presented earlier followed by practice where needed. Van Engen (102) noted implications of present knowledge for future trends in teaching arithmetic.

It is apparent from this review that research in arithmetic has changed from consideration of mechanics of computation to concern for problems of learning and concept formation. Future research might profitably be done by teams of teachers and experts in arithmetic working together to discover more about how children learn arithmetic.

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The Natural Sciences and Mathematics || Mathematics in the Elementary Grades